Subgroups — #27277212

Order 13568.

418 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810753	C₂	1	1	yes	maximal, characteristic
2	#20274843	C₂	1	1	yes	maximal, characteristic
3	#20274721	C₂	1	2	yes	maximal
4	#20272599	C₂	1	2	yes	maximal
5	C₂×C₅₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
7	#16810753	—	2	2	no	—
8	#16810753	C₄	1	4	yes	complemented
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	C₂×#13133146	—	2	1	no	—
11	C₂×#13133142	—	2	2	no	—
12	C₂×#13133127	—	2	1	no	—
13	#16810493	C₂²	1	2	yes	—
14	#16810386	—	2	2	no	—
15	#16810386	—	2	1	no	—
16	#16810386	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#334	—	2	2	no	—
20	C₂²×Dic₁₀₆	D₄	1	1	yes	characteristic
21	C₂²×#88322	—	2	1	no	—
22	C₄×Dic₁₀₆	—	2	1	no	—
23	C₄×Dic₁₀₆	—	2	1	no	—
24	C₄×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₆	Q₈	1	2	yes	—
26	C₂×C₄⋊Dic₅₃	—	2	2	no	—
27	C₂×C₄⋊Dic₅₃	—	2	2	no	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	D₄	1	1	yes	characteristic
30	C₂×#11719919	—	2	1	no	—
31	C₂×C₄×Dic₅₃	—	2	1	no	—
32	C₄⋊#88322	—	2	1	no	—
33	C₄⋊#88322	C₂×C₄	1	2	yes	—
34	#13133146	—	2	4	no	—
35	#13133142	—	4	4	no	—
36	#13133127	—	2	4	no	—
37	C₂²×C₈×C₅₃	—	2	2	no	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₅₃	D₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
48	C₂²×Dic₅₃	—	2	1	no	—
49	C₂²×Dic₅₃	—	2	1	no	—
50	C₂²×Dic₅₃	—	4	2	no	—
51	C₂×Dic₁₀₆	—	2	2	no	—
52	C₂×Dic₁₀₆	—	2	2	no	—
53	C₂×Dic₁₀₆	—	2	1	no	—
54	C₂×Dic₁₀₆	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₅₃	#41	1	1	yes	characteristic
66	C₄⋊Dic₅₃	—	2	1	no	—
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	4	2	no	—
69	C₄⋊Dic₅₃	—	4	2	no	—
70	C₄⋊Dic₅₃	—	4	2	no	—
71	C₄⋊Dic₅₃	—	8	2	no	—
72	C₄⋊Dic₅₃	C₄⋊C₄	1	2	yes	—
73	#11719919	—	2	2	no	—
74	#11719919	—	2	2	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	C₄×Dic₅₃	—	2	2	no	—
77	C₂×#88322	—	2	2	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂³×C₅₃	#121	1	1	yes	characteristic
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#135	1	2	yes	—
88	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×C₄×C₅₃	#137	1	2	yes	—
90	C₂×C₄×C₅₃	#137	1	2	yes	—
91	C₂×Dic₅₃	—	2	2	no	—
92	C₂×Dic₅₃	—	2	2	no	—
93	C₂×Dic₅₃	—	4	1	no	—
94	C₂×Dic₅₃	—	4	1	no	—
95	C₂×Dic₅₃	—	4	2	no	—
96	C₂×Dic₅₃	—	8	2	no	—
97	C₂×Dic₅₃	—	8	2	no	—
98	C₂×Dic₅₃	—	8	2	no	—
99	Dic₁₀₆	—	2	4	no	—
100	Dic₁₀₆	—	4	2	no	—
101	C₈×C₅₃	—	2	2	no	—
102	C₈×C₅₃	—	2	2	no	—
103	C₈×C₅₃	—	8	2	no	—
104	#88322	—	4	2	no	—
105	#8657	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	#380	1	1	yes	characteristic
107	C₂²×C₅₃	#464	1	2	yes	—
108	C₂²×C₅₃	#469	1	1	yes	characteristic
109	C₂²×C₅₃	#470	1	1	yes	characteristic
110	C₂²×C₅₃	#471	1	2	yes	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	2	1	no	—
113	C₄×C₅₃	#416	1	1	yes	characteristic
114	C₄×C₅₃	#419	1	1	yes	characteristic
115	C₄×C₅₃	#428	1	2	yes	—
116	C₄×C₅₃	#468	1	2	yes	—
117	C₄×C₅₃	#479	1	2	yes	—
118	Dic₅₃	—	4	2	no	—
119	Dic₅₃	—	4	2	no	—
120	Dic₅₃	—	16	2	no	—
121	C₂×#446	—	53	1	no	—
122	#1834	—	53	1	no	—
123	#1803	—	53	2	no	—
124	C₂×#334	D₅₃	1	1	yes	characteristic
125	#1296	—	53	2	no	—
126	C₂×C₅₃	#1702	1	2	yes	—
127	C₂×C₅₃	#1704	1	2	yes	—
128	C₂×C₅₃	#1805	1	1	yes	characteristic
129	C₂×C₅₃	#1809	1	1	yes	characteristic
130	C₂×C₅₃	#1815	1	1	yes	characteristic
131	C₂×C₄×Q₈	—	53	1	no	—
132	#446	—	53	4	no	—
133	#446	—	106	2	no	—
134	C₂×C₄⋊C₈	—	53	1	no	—
135	C₂×C₄⋊C₈	—	106	1	no	—
136	C₂×C₄⋊C₈	C₂×D₅₃	1	2	yes	—
137	C₂×#103	—	106	2	no	—
138	C₂×#97	—	106	1	no	—
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	#384	—	53	2	no	—
141	#340	—	106	2	no	—
142	#340	—	106	1	no	—
143	#340	—	106	1	no	—
144	#334	—	2	2	no	—
145	C₅₃	#8657	1	1	yes	characteristic, complemented, Sylow
146	C₂²×Q₈	—	53	1	no	—
147	C₂²×C₈	—	2	2	no	—
148	C₂²×C₈	—	106	1	no	—
149	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
150	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
151	C₄×Q₈	—	53	2	no	—
152	C₄×Q₈	—	53	2	no	—
153	C₄×Q₈	—	106	1	no	—
154	C₄×Q₈	—	106	1	no	—
155	C₂×C₄⋊C₄	—	53	1	no	—
156	C₂×C₄⋊C₄	—	106	1	no	—
157	C₂×C₄⋊C₄	—	106	2	no	—
158	C₂×C₄⋊C₄	—	106	2	no	—
159	C₂×C₄⋊C₄	—	106	2	no	—
160	C₂×C₄²	—	106	1	no	—
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	C₄⋊C₈	—	53	2	no	—
165	C₄⋊C₈	—	106	4	no	—
166	C₄⋊C₈	—	106	1	no	—
167	#103	—	212	4	no	—
168	#97	—	106	4	no	—
169	C₄×C₈	C₄×D₅₃	1	2	yes	—
170	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
171	C₂×Q₈	—	53	2	no	—
172	C₂×Q₈	—	106	1	no	—
173	C₂×Q₈	—	106	2	no	—
174	C₂×Q₈	—	106	2	no	—
175	C₂²×C₄	—	106	1	no	—
176	C₂²×C₄	—	106	1	no	—
177	C₂²×C₄	—	212	2	no	—
178	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
179	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
180	C₂²×C₄	#11719949	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	106	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	#11719921	1	2	yes	—
189	C₂×C₈	#11719921	1	2	yes	—
190	C₄⋊C₄	—	53	2	no	—
191	C₄⋊C₄	—	106	2	no	—
192	C₄⋊C₄	—	106	1	no	—
193	C₄⋊C₄	—	106	2	no	—
194	C₄⋊C₄	—	424	2	no	—
195	C₄⋊C₄	—	212	2	no	—
196	C₄⋊C₄	—	212	2	no	—
197	C₄⋊C₄	—	212	2	no	—
198	C₄⋊C₄	—	212	2	no	—
199	C₄²	—	106	2	no	—
200	C₄²	—	106	2	no	—
201	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
202	C₄²	#11719947	1	1	yes	characteristic
203	C₄²	Q₈×D₅₃	1	1	yes	characteristic
204	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
205	C₂³	#13133192	1	1	yes	characteristic
206	Q₈	—	106	4	no	—
207	Q₈	—	212	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	106	2	no	—
210	C₂×C₄	—	106	2	no	—
211	C₂×C₄	—	424	2	no	—
212	C₂×C₄	—	424	2	no	—
213	C₂×C₄	—	424	2	no	—
214	C₂×C₄	—	212	1	no	—
215	C₂×C₄	—	212	1	no	—
216	C₂×C₄	—	212	2	no	—
217	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
218	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄	#13133175	1	1	yes	characteristic
220	C₂×C₄	#13133188	1	1	yes	characteristic
221	C₂×C₄	#13133193	1	1	yes	characteristic
222	C₂×C₄	#13133197	1	1	yes	characteristic
223	C₂×C₄	#13133239	1	2	yes	—
224	C₂×C₄	#13133244	1	2	yes	—
225	C₂×C₄	#13133246	1	2	yes	—
226	C₂×C₄	#13133250	1	2	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	8	2	no	—
230	C₈	—	212	2	no	—
231	C₂²	#16810480	1	1	yes	characteristic
232	C₂²	#16810846	1	2	yes	—
233	C₂²	#16810877	1	1	yes	characteristic
234	C₂²	#16810882	1	1	yes	characteristic
235	C₂²	#16810887	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	848	2	no	—
239	C₄	—	212	2	no	—
240	C₄	—	212	2	no	—
241	C₄	#16810620	1	1	yes	characteristic
242	C₄	#16810633	1	1	yes	characteristic
243	C₄	#16810662	1	2	yes	—
244	C₄	#16810866	1	2	yes	—
245	C₄	#16810931	1	2	yes	—
246	C₂	#20274284	1	2	yes	—
247	C₂	#20274298	1	2	yes	—
248	C₂	#20274731	1	1	yes	characteristic
249	C₂	#20274745	1	1	yes	characteristic
250	C₂	#20274771	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133192
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₅₃×#334	C₂
socle, prime radical	C₂³×C₅₃	#121
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#334	D₅₃
hypocenter	C₅₃	#8657
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#334

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
8. #16810753	C₄	semidirect
145. C₅₃	#8657	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16810753	2
#20274843	2
#20274721	2
#20272599	2
C₂×C₅₃×#334	2
#8657	53